On Semifields of Type

I I G ◭◭ ◮◮ ◭ ◮ On semifields of type (q2n,qn,q2,q2,q), n odd page 1 / 19 ∗ go back Giuseppe Marino Olga Polverino Rocco Tromebtti full screen Abstract 2n n 2 2 close A semifield of type (q , q , q , q , q) (with n > 1) is a finite semifield of order q2n (q a prime power) with left nucleus of order qn, right 2 quit and middle nuclei both of order q and center of order q. Semifields of type (q6, q3, q2, q2, q) have been completely classified by the authors and N. L. Johnson in [10]. In this paper we determine, up to isotopy, the form n n of any semifield of type (q2 , q , q2, q2, q) when n is an odd integer, proving n−1 that there exist 2 non isotopic potential families of semifields of this type. Also, we provide, with the aid of the computer, new examples of semifields of type (q14, q7, q2, q2, q), when q = 2. Keywords: semifield, isotopy, linear set MSC 2000: 51A40, 51E20, 12K10 1. Introduction A finite semifield S is a finite algebraic structure satisfying all the axioms for a skew field except (possibly) associativity. The subsets Nl = {a ∈ S | (ab)c = a(bc), ∀b, c ∈ S} , Nm = {b ∈ S | (ab)c = a(bc), ∀a, c ∈ S} , Nr = {c ∈ S | (ab)c = a(bc), ∀a, b ∈ S} and K = {a ∈ Nl ∩ Nm ∩ Nr | ab = ba, ∀b ∈ S} are fields and are known, respectively, as the left nucleus, the middle nucleus, the right nucleus and the center of the semifield. A finite semifield is a vector space ∗ ACADEMIA This work was supported by the Research Project of MIUR (Italian Office for University and Re- PRESS search) “Strutture geometriche, combinatoria e loro applicazioni” and by INDAM (Istituto Nazionale di Alta Matematica “Francesco Severi”. I I G over its nuclei and its center (for more details on semifields see e.g. [4, 9]). If S ◭◭ ◮◮ satisfies all the axioms of a semifield except possibly the existence of the identity element of the multiplication, then S is called pre-semifield. From now on the ◭ ◮ terms semifield and pre-semifield will be always used to denote a finite semifield and a finite pre-semifield. If S S is a pre-semifield, then S is an page 2 / 19 = ( , +, ◦) ( , +) elementary abelian p-group (p prime); hence, S is an Fp-vector space. Now, if S S S′ S′ ′ go back = ( , +, ◦) and = ( , +, ◦ ) are two pre-semifields whose additive groups are elementary abelian p-groups, then S and S′ are isotopic if there exist three F S S′ full screen invertible p-linear maps, f1, f2 and f3 of into such that ′ f1(x) ◦ f2(y) = f3(x ◦ y) , close for each x, y ∈ S. The dimensions of a semifield S over its nuclei and its center quit are invariant under the isotopy relation. From any pre-semifield it is possible to construct a semifield which is isotopic to the starting pre-semifield. The sizes of the nuclei as well as the size of the center of a semifield are invariant under isotopy. Semifields coordinatize certain translation planes (called semifield planes) and two semifield planes are isomorphic if and only if the corresponding semifields are isotopic (see [1]). A semifield is isotopic to a field if and only if the corresponding semifield plane is Desarguesian. Let b be an element of a semifield S with center K; then the map ϕb mapping x ∈ S to xb ∈ S is a linear map when S is regarded as a left vector space over Nl. The set S = {ϕb | b ∈ S} is called the spread set of linear maps of S; it is closed under the sum of linear maps, |S| = |S| and λϕb = ϕλb for any λ ∈ K, i.e. S is a K-vector subspace of the vector space V of all Nl-linear maps of S. We say that a semifield is of type (q2n, qn, q2, q2, q) (q a prime power), if it has order q2n, left nucleus of order qn, right and middle nuclei both of order q2 and center of order q. In this paper we prove that a semifield S of type (q2n, qn, q2, q2, q), n odd, is n+1 S F 2 isotopic to a semifield j =( q n , +, ◦), 2 ≤ j < n, with multiplication given by qn x ◦ y =(α1 + α2a2 + ··· + αjaj)x +(β1b1 + β2b2 + ··· + βn−jbn−j)x , where y = α1 + α2a2 + ··· + αjaj + β1b1 + β2b2 + ··· + βn−jbn−j (αi, βi ∈ Fq2 ) and {1, a2,...,aj, b1, b2,...,bn−j} is an Fq2 -basis of Fq2n such that n α + α a + ··· + α a q +1 1 2 2 j j 6= 1 , β b + β b + ··· + β − b − 1 1 2 2 n j n j ACADEMIA F PRESS for each αi, βi ∈ q2 , (β1,...,βn−j) 6= (0,..., 0). Moreover, by using the geo- metric properties of the spread sets of linear maps associated with such semi- ′ ′ S S ′ n+1 fields we prove that two semifields j and j with 2 ≤ j, j < n and j 6= j I I G are not isotopic. Hence the semifields of type (q2n, qn, q2, q2, q), n odd, are par- ◭◭ ◮◮ n−1 titioned into 2 non-isotopic families. ◭ ◮ Some semifields of type S n+1 are constructed by the authors and N. Johnson 2 in [11, Section 4] generalizing the cyclic semifields [8]. Moreover, semifields page 3 / 19 of type Sn−1 belong to a family of semifields introduced in [16] and studied in [20]. The other classes seem to be likely places to search for new semifields. go back Indeed, computational results obtained using MAGMA provide some new examples of semifields of type (q14, q7, q2, q2, q), for q = 2 and j = 5. full screen close 2. Preliminary results quit Let S = (Fq2n , +, ◦) be a semifield two dimensional over its left nucleus Fqn , with identity element 1 and center Fq, and let S be the spread set of Fqn -linear maps of Fq2n defining the multiplication ◦, i.e. x ◦ y = ϕy(x) where ϕy is the unique element of S such that ϕy(1) = y. Since S has center Fq, we have x ◦ y = xy = y ◦ x for each x ∈ Fq and y ∈ Fq2n , and hence S contains the field of linear maps Fq = {x ∈ Fq2n 7→ αx ∈ Fq2n | α ∈ Fq}. An element z ∈ Fq2n belongs to the right nucleus of S if x ◦ (y ◦ z)=(x ◦ y) ◦ z for each x, y ∈ Fq2n , i.e. z ∈ Nr if ϕy◦z = ϕzϕy (juxtaposition stands for the composition of maps) for each y ∈ Fq2n . So the right nucleus of S defines a field Nr = {ϕz | z ∈ Nr} of linear maps contained in S isomorphic to Nr with respect to which S is a left vector space, i.e. NrS = {µϕ | µ ∈ Nr, ϕ ∈ S} = S and Nr can be characterized as the maximal field of linear maps contained in S with respect to which S is a left vector space. Similarly, the middle nucleus of S defines a field Nm = {ϕz | z ∈ Nm} of linear maps contained in S isomorphic to Nm with respect to which S is a right vector space, i.e. SNm = S and Nm can be characterized as the maximal field of linear maps contained in S with respect to which S is a right vector space. If Ψ and Φ are invertible Fqn -linear maps of Fq2n and σ is an automorphism of Fq2n , then the set ′ S = ΨSσΦ = {ΨϕσΦ | ϕ ∈ S} (∗) n n (where ϕσ : x 7→ aσx + bσxq for ϕ : x 7→ ax + bxq and the composition of maps is to be read from right to left) is an additive spread set of linear maps that ′ defines on Fq2n a pre-semifield S isotopic to S. Conversely, any pre-semifield ′ ′ ′ S = (Fq2n , +, ◦ ) isotopic to S is defined by a spread set S of type (∗) (see ACADEMIA e.g. [10, 11]). PRESS σ −1 N ′ Note that ΨNr Ψ is a field, isomorphic to r, with respect to which S σ −1 ′ ′ −1 σ is a left vector space, i.e. ΨNr Ψ S = S and similarly Φ NmΦ is a field, I I G N ′ −1 σ ′ isomorphic to m, such that S Φ NmΦ = S . Hence we have the following ◭◭ ◮◮ property. ′ ′ ◭ ◮ Property 2.1. If S is a pre-semifield, with associated spread set S , isotopic to the semifield S, then the right (respectively, middle) nucleus of S is isomorphic to the page 4 / 19 maximal field K of linear maps contained in V with respect to which KS′ = S′ (respectively, S′K = S′). go back Any element ϕ ∈ V can be uniquely written as full screen qn ϕ = ϕa,b : x ∈ Fq2n 7→ ax + bx ∈ Fq2n close qn+1 qn+1 and ϕa,b is non-invertible if and only if a = b (see [14, p. 361]). Since qn+1 qn+1 q(ϕa,b) = a − b is a quadratic form of V over Fqn , the non-invertible quit elements of V define the hyperbolic quadric qn+1 qn+1 Q = [ϕa,b]Fqn | a − b = 0, (a, b) 6= (0, 0) n o n of the 3-dimensional projective space P = PG(V, Fqn ) = PG(3, q ). Here the F n V symbol [ϕa,b]Fqn denotes the 1-dimensional q -vector subspace of generated by ϕa,b. If S is the spread set of Fqn -linear maps of a pre-semifield S = (Fq2n , +, ◦) with center Fq, then S is an Fq-vector subspace of V of dimension 2n and all the non-zero elements of S are invertible maps.

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